/**
* @brief Covariance matrix between the functional and derivative training data
* or its partial derivative
* given pair-wise squared distances and differences
* @param [in] logHyp The log hyperparameters
* - logHyp(0) = \f$\log(l)\f$
* - logHyp(1) = \f$\log(\sigma_f)\f$
* @param [in] pSqDist The pair-wise squared distances between the functional and derivative training data
* @param [in] pDelta The pair-wise differences between the functional and derivative training data
* @param [in] pdHypIndex (Optional) Hyperparameter index for partial derivatives
* - pdHypIndex = -1: return \f$\frac{\partial \mathbf{K}(\mathbf{X}, \mathbf{Z})}{\partial \mathbf{Z}_j}\f$ (default)
* - pdHypIndex = 0: return \f$\frac{\partial^2 \mathbf{K}(\mathbf{X}, \mathbf{Z})}{\partial \log(l) \partial \mathbf{Z}_j}\f$
* - pdHypIndex = 1: return \f$\frac{\partial^2 \mathbf{K}(\mathbf{X}, \mathbf{Z})}{\partial \log(\sigma_f) \partial \mathbf{Z}_j}\f$
* @return An NNxNN matrix pointer\n
* NN: The number of functional and derivative training data
*/
static MatrixPtr K_FD(const Hyp &logHyp,
const MatrixConstPtr pSqDist,
const MatrixConstPtr pDelta,
const int pdHypIndex = -1)
{
// K: same size with the squared distances
MatrixPtr pK = K(logHyp, pSqDist);
// constants
const Scalar inv_ell2 = exp(static_cast<Scalar>(-2.f) * logHyp(0)); // (1/ell^2)
// pre-calculation
// k(x, z) = sigma_f^2 * exp(-r^2/(2*ell^2)), r = |x-z|
// k(s) = sigma_f^2 * exp(s), s = -r^2/(2*ell^2)
//
// s = -r^2/(2*ell^2) = (-1/(2*ell^2)) * sum_{i=1}^d (xi - zi)^2
// ds/dzj = (xj - zj) / ell^2
//
// dk/ds = sigma_f^2 * exp(s) = k
// dk(s)/dzj = dk/ds * ds/dzj
// = k(x, z) * (xj - zj) / ell^2
pK->noalias() = (inv_ell2 * pK->array() * pDelta->array()).matrix();
// mode
switch(pdHypIndex)
{
// derivatives of covariance matrix w.r.t log ell
case 0:
{
// dk/dzj = sigma_f^2 * exp(s) * (xj - zj) / ell^2
// d^2k/dzj dlog(ell) = sigma_f^2 * exp(s) * [(xj - zj) / ell^2] * (r^2/ell^2 - 2)
// = dk/dzj * (r^2/ell^2 - 2)
pK->noalias() = (pK->array() * (inv_ell2 * (pSqDist->array()) - static_cast<Scalar>(2.f))).matrix();
break;
}
// derivatives of covariance matrix w.r.t log sigma_f
case 1:
{
// d^2k/dzj dlog(sigma_f) = 2 * dk/dzj
pK->noalias() = static_cast<Scalar>(2.f) * (*pK);
break;
}
// covariance matrix, dk/dzj
default:
{
break;
}
}
return pK;
}
/**
* @brief Covariance matrix between the derivative training data
* or its partial derivative
* given pair-wise squared distances and differences
* @param [in] logHyp The log hyperparameters
* - logHyp(0) = \f$\log(l)\f$
* - logHyp(1) = \f$\log(\sigma_f)\f$
* @param [in] pSqDist The pair-wise squared distances between the functional and derivative training data
* @param [in] pDelta_i The pair-wise differences of the i-th components between the functional and derivative training data
* @param [in] pDelta_j The pair-wise differences of the j-th components between the functional and derivative training data
* @param [in] fSameCoord i == j
* @param [in] pdHypIndex (Optional) Hyperparameter index for partial derivatives
* - pdHypIndex = -1: return \f$\frac{\partial^2 \mathbf{K}(\mathbf{X}, \mathbf{Z})}{\partial \mathbf{X}_i \partial \mathbf{Z}_j}\f$ (default)
* - pdHypIndex = 0: return \f$\frac{\partial^3 \mathbf{K}(\mathbf{X}, \mathbf{Z})}{\partial \log(l) \partial \mathbf{X}_i \partial \mathbf{Z}_j}\f$
* - pdHypIndex = 1: return \f$\frac{\partial^3 \mathbf{K}(\mathbf{X}, \mathbf{Z})}{\partial \log(\sigma_f) \partial \mathbf{X}_i \partial \mathbf{Z}_j}\f$
* @return An NNxNN matrix pointer\n
* NN: The number of functional and derivative training data
*/
static MatrixPtr K_DD(const Hyp &logHyp,
const MatrixConstPtr pSqDist,
const MatrixConstPtr pDelta_i,
const MatrixConstPtr pDelta_j,
const bool fSameCoord,
const int pdHypIndex = -1)
{
// K: same size with the squared distances
MatrixPtr pK = K(logHyp, pSqDist, pdHypIndex);
// hyperparameters
const Scalar inv_ell2 = exp(static_cast<Scalar>(-2.f) * logHyp(0)); // 1/ell^2
const Scalar inv_ell4 = inv_ell2 * inv_ell2; // 1/ell^4
const Scalar four_inv_ell4 = static_cast<Scalar>(4.f) * inv_ell4; // 4/ell^4
// delta
const Scalar delta_inv_ell2 = fSameCoord ? inv_ell2 : static_cast<Scalar>(0.f); // delta(i, j)/ell^2
const Scalar neg_double_delta_inv_ell2 = static_cast<Scalar>(-2.f) * delta_inv_ell2; // -2*delta(i, j)/ell^2
// pre-calculation
// k(x, z) = sigma_f^2 * exp(-r^2/(2*ell^2)), r = |x-z|
// k(s) = sigma_f^2 * exp(s), s = -r^2/(2*ell^2)
//
// s = -r^2/(2*ell^2) = (-1/(2*ell^2)) * sum_{i=1}^d (xi - zi)^2
// ds/dzj = (xj - zj) / ell^2
//
// dk/ds = sigma_f^2 * exp(s) = k
// dk(s)/dzj = dk/ds * ds/dzj
// = k(x, z) * (xj - zj) / ell^2
//
// d^2k/dzj dxi = dk/dxi * (xj - zj) / ell^2 + k * delta(i, j) / ell^2
// = k * [-(xi - zi)/ell^2] * (xj - zj) / ell^2 + k * delta(i, j) / ell^2
// = k * [delta(i, j)/ell^2 - (xi - zi)*(xj - zj)/ell^4]
pK->noalias() = (pK->array() * (delta_inv_ell2 - inv_ell4 * pDelta_i->array() * pDelta_j->array())).matrix();
// particularly, derivatives of covariance matrix w.r.t log ell
if(pdHypIndex == 0)
{
// d^2k/dxi dzj = k * [delta(i, j)/ell^2 - (xi - zi)*(xj - zj)/ell^4]
// d^3k/dxi dzj dlog(ell) = dk/dlog(ell) * [delta(i, j)/ell^2 - (xi - zi)*(xj - zj)/ell^4]
// + k * [-2*delta(i, j)/ell^2 + 4*(xi - zi)*(xj - zj)/ell^4]
MatrixPtr pK0 = K(logHyp, pSqDist);
pK->noalias() += (pK0->array() * (neg_double_delta_inv_ell2 + four_inv_ell4 * pDelta_i->array() * pDelta_j->array())).matrix();
}
return pK;
}
